Nicholson-Bailey Model

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The Nicholson-Bailey model developed by Alexander Nicholson and Victor Bailey uses difference equations to simulate the population dynamics of the parasitoid-host relationship.

Background

The life cycle of an aphid parasitoid

Parasitoidism is a relationship classified between predation and parasitism. Predator-prey dynamics have low "host" intimacy but high "host" lethality, and parasite-host dynamics have high intimacy but low lethality. Parasitoids combine these, having both high intimacy and high lethality. The parasitoid-host relationship is unique to the insect world, to which it is well suited. Insects typically have short life spans and correspondingly quick reproductive rates that are optimal for parasitoidism.

The parasitoid life cycle follows the pattern of complete metamorphosis -- meaning there are four stages: egg, larva, pupa, and adult. Parasitoids infect hosts by laying eggs inside the host. The eggs hatch into larvae, which then eat the host alive. When the host dies the larvae pupate, and eventually emerge from the host body as adults.

Nicholson, a biologist, was interested to see if the parasitoid-host relationship could be modeled, and thus predicted, mathematically. The parasitoid-host relationship is a dynamic one that is dependent on many factors. Modeling the relationship allows a better understanding of how parasitoids and hosts behave under different circumstances.

Assumptions

Before modeling the parasitoid-host relationship, Nicholson and Bailey had to make certain assumptions about the parasitoids and their prey. They assumed that all infected hosts would produce a new generation of parasitoids, whereas all uninfected hosts would have their own offspring. Nicholson and Bailey also assumed that they could assign a function to represent the random encounter of the homogenously mixed two species that would result in the infection of the hosts. They also made clear that each host could only be infected once by a parasitoid.

The Initial Model

Nicholson and Bailey developed the following linear difference equations to model the parasitoid-host relationship:

Nt+1 = λNt e-aPt
Pt+1 = cNt[1 - e-aPt]
Where
Nt = density of host species in generation t
Pt = density of parasitoid in generation t
a = searching efficiency of parasitoids
λ = host reproductive rate
c = average number of viable eggs laid by a parasitoid on a single host.

The equation for the generational population of the host is dependent on the host’s reproductive rate and the population of the previous generation that had not been parasitized. On the other side of the relationship, the equation for the generational population of the parasitoids is dependent on the number of eggs laid on the number of infected hosts of the previous generation. Both equations include the term e-aPt, a measure of the fraction of hosts not parasitized.

Initial Results

Results of the initial Nicholson-Bailey model. Note that the oscillations are unstable and eventually both populations crash.

With their model, Nicholson and Bailey were able to manipulate their equations to investigate when the populations of the predators and prey were at steady state, or not changing. Intuitively, they found that if the hosts had a birth rate under a certain level that the hosts would become extinct. Consequently, with no hosts to prey on, the parasitoids would go extinct as well. This extinction steady state is common in many biological systems. Nicholson and Bailey were also able to derive a non-trivial steady state, but it was unstable and thus not very helpful.

Example.jpg

Systems like these that model interactions between species and other dynamic phenomena need steady states to actually be useful. If steady states exist and are stable, no matter where the two-species are in the system, they will tend towards these states. Although the Nicholson-Bailey model has such levels or states, they are not stable. The model shows that slight deviations from the steady state create some really weird and chaotic results which are not entirely feasible. This in no way disqualifies the model from being effective. It just goes to show that there are some imperfections in the model and that measures should be taken to strengthen the model so as to make it more realistic. The argument about the practicality of these states arises from the existent of two-species interactions that have states that are evidently more stable than what the Nicholson-Bailey model forecasts.

Improvements

The Nicholson-Bailey model was able to accurately predict parasitoid-host populations of insects in a laboratory setting. In this regard, the model was a success. Despite this, the fact that the model was not able to predict non-extinction population stability was disappointing. Prediction of the circumstances that lead to population stability is necessary for the biological analysis of this any many other systems. Because of this, it was necessary to improve the model to allow for it to predict population stability. They did this by replacing the host reproductive rate λ with the function er(1-Nt/K) where K is the carrying capacity of hosts in the system; this alteration created a density dependence of the parasitoid and host populations on the capacity of the system. The assumption was made that without parasitoids, the host population would eventually reach this carrying capacity (as opposed to growing exponentially as in the initial model).

Improved model with Density Dependence:

Nt+1 = Nter(1-Nt/K)-aPt
Pt+1 = cNt[1 - e-aPt]

N-b steady states.bmp

This change to the model produced a steady state (in addition to the extinction one) where stability was possible. Depending on the parameters of the system, this steady state could be a stable spiral, a stable orbit (limit cycle), a period 5 orbit, or chaotic. A Bifurcation Diagram can be used to evaluate what happens to the stability of the steady state for different parameter ranges.

Some other improvements relating how the hosts are spread out and how this affects the rate of attack can also be used to improve the system. Assuming that hosts are not spread out evenly within an area will help make this model a slightly more realistic. In the model, there is an underlying assumption of a uniform distribution of hosts and as such each host has the same chance of being attacked in a habitat. That should not be the case. There are areas within the habitat that are densely populated with hosts and others that are not and so the parasitoids will tend to attack hosts in these densely populated areas more than those in less populated areas. This will help bring out the disproportion existent in the likelihood of hosts being attacked. Also, there should be a cap on how densely populated the habitat can be. This is necessary because there is a definite capacity (maximum number of elements the habitat can handle before things get messy), and as such that has to be factored into the calculations. Putting that in there will impact a lot of the different variables in the system and will also make the system a little more realistic as the variables will have a bound within which they can vary.

Related Topics

Lotka-Volterra Model -- Predator-prey model

Logistic Growth Model -- Population growth with carrying capacity